Gateaux directional differentiability is in general too weak of a
differentiability concept to ensure the validity of the functional
delta method. However, when the map \(\phi \)  is also locally Lipschitz
at \(x\in X_{D}\)  (in the sense that there exists a \(C>0\)  such that
for all \(x^{\prime },x^{\prime \prime }\in X_{D}\)  in some neighborhood of \(x\) , \(\Vert \phi (x^{\prime })-\phi (x^{\prime \prime })\Vert _{Y}\le C\Vert x^{\prime }-x^{\prime \prime }\Vert _{X}\) ),
Gateaux and Hadamard directional differentiability become equivalent
([1]}). This is convenient and
holds for the maps considered in the next lemma. In this lemma, we
let \(\Gamma \)  and \(\Delta \)  denote any compact and non-empty Euclidean
sets (which need not equal the sets \(\Gamma \)  and \(\Delta \)  elsewhere
in the paper; the same notation is used for convenience). Furthermore, for
any function \(x\in C(\Delta )\)  we denote \(\Delta _{0}(x)=\lbrace \delta _{0}\in \Delta :x(\delta _{0})=\sup _{\delta \in \Delta }x(\delta )\rbrace \) ,
and for any function \(x\in C(\Gamma \times \Delta )\)  we let \(\bar{x}(\gamma )=\sup _{\delta \in \Delta }x(\gamma ,\delta )\) 
denote the max-function and define the sets
\(\Theta _{0}(x) & =\lbrace (\gamma _{0},\delta _{0})\in \Gamma \times \Delta :x(\gamma _{0},\delta _{0})=\sup _{\delta \in \Delta }x(\gamma _{0},\delta )=\bar{x}(\gamma _{0})\text{ and }\bar{x}(\gamma _{0})=\inf _{\gamma \in \Gamma }\bar{x}(\gamma )\rbrace ,\\\Gamma _{0}(x) & =\lbrace \gamma _{0}\in \Gamma :(\gamma _{0},\delta _{0})\in \Theta _{0}(x)\text{ for some }\delta _{0}\in \Delta \rbrace ,\vphantom{\sup _{\delta \in \Delta }}\\\Delta _{0}(\gamma ,x) & =\lbrace \delta _{0}\in \Delta :x(\gamma ,\delta _{0})=\sup _{\delta \in \Delta }x(\gamma ,\delta )\rbrace \quad (\gamma \in \Gamma ).\) 
